The Doppler shift phenomenon is well understood when the source and observer are in relative constant motion. However, I'm curious to know how the Doppler shift phenomenon is modified when they (i.e., the source and the observer) are in relative accelerating motion (any type of accelerating motion, e.g., linear acceleration, circular motion, etc.). Additionally, how the speed of the wave, generated by the source and moving towards the observer, is affected by the nature of non-inertial frames?
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$\begingroup$ If the transmitted EM signal is sufficiently wide bandwidth then the received pulse envelope will be distorted and not just the carrier frequency being shifted when for $k>1$ and the $k^{th}$ order range rate $\frac{d^kR(t)}{dt^k}\ne 0$ , $R$ being the range. For details see Rihaczek: High Resolution Radar, section 3.3. $\endgroup$– hyportnexCommented Mar 12 at 0:30
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Provided spacetime is flat, even if the emitter and the receiver are accelerating, you can still analyze the scenario in an inertial frame. The usual Doppler shift formula applies between the velocity of the emitter at the time of emission and the velocity of the receiver at the time of reception.
If spacetime is not flat then there is no one easy formula. You have to parallel transport a null vector along a null geodesic connecting the emission and the reception events. The contraction of the four-velocity of the emitter with the emission null vector gives the frequency at emission. Similarly the contraction of the four velocity of the receiver with the reception null vector gives the frequency at reception. The ratio is the Doppler shift.
The curved spacetime approach also works in non-inertial frames in flat spacetime. However, it is not usually helpful, even when analyzing non-inertial objects.
